This note studies the inhomogeneous generalized Hartree equation

The goal of this work is two-fold. First, one obtains the existence of a local solution in \(C_T(H^{s_c})\), where the critical Sobolev exponent is given by the equality \(\lambda ^\frac{2-2\rho +\gamma }{2(p-1)}\Vert u_0(\lambda \cdot )\Vert _{\dot{H}^{s_c}}=\Vert u_0\Vert _{\dot{H}^{s_c}}\). Second, one investigates the uniqueness of critical solutions in \(C_T(H^{s_c})\). Indeed, since one uses a fixed point argument in some Strichartz spaces, the uniqueness in the energy space is not trivial. In fact, the technique used in order to obtain the existence of a local sub-critical solution, which consists to divide the integrals on the unit centered ball of \(\mathbb {R}^N\) and it’s complementary is no more sufficient to conclude. To overcome this difficulty, one uses a fixed point argument with Strichartz estimates in some suitable Lorentz spaces, which enables us to handle the inhomogeneous term by the fact that \(|x|^{-\rho }\in L^{\frac{N}{\rho },\infty }\).

本文研究非齐次广义 Hartree 方程

这项工作的目标有两个。首先，获得\(C_T(H^{s_c})\)中局部解的存在性，其中临界 Sobolev 指数由等式\(\lambda ^\frac{2-2\rho +\gamma }{2(p-1)}\Vert u_0(\lambda \cdot )\Vert _{\dot{H}^{s_c}}=\Vert u_0\Vert _{\dot{H}^{s_c}} \)。其次，研究\(C_T(H^{s_c})\)中关键解决方案的唯一性。事实上，由于在某些 Strichartz 空间中使用不动点参数，因此能量空间中的唯一性并非微不足道。事实上，为了获得局部亚临界解的存在性而使用的技术，即除以\(\mathbb {R}^N\)的单位中心球上的积分及其互补，已经不够了得出结论。为了克服这一困难，我们在一些合适的洛伦兹空间中使用带有 Strichartz 估计的不动点参数，这使我们能够通过以下事实来处理非齐次项：\(|x|^{-\rho }\in L^{\frac {N}{\rho },\infty }\)。